Energy Basilio Bona DAUIN - Politecnico di Torino October 2013 Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 1 / 54
Introduction Analytical Approach The multibody system is considered as a system in which the dynamic equations derive from a unifying principle . This principle is based on the fact that, in order to describe the motion of a system, it is sufficient to consider some scalar quantities. These were in origin called vis viva and work function , nowadays are called kinetic energy and potential energy . Both are state functions , i.e., those functions that map the value of the state vector into a scalar function. The concept of state will be defined later; for the moment we simply consider that the state corresponds to the two vectors q ( t ) and ˙ q ( t ). Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 2 / 54
This general principle is the principle of least action . Let us consider the space Q of the generalized coordinates q ∈ Q , as sketched in Figure for a two-dimensional space Q . A particle starts its motion at time t 1 in Q 1 = q ( t 1 ) and ends it motion at time t 2 reaching the state Q 2 = q ( t 2 ). Assume that the motion keeps constant the sum E = C + P of the kinetic energy C and the potential energy P that the particle has at time t 1 . Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 3 / 54
Q 1 and Q 2 are connected by a continuous path (trajectory) called true trajectory , and is unknown, since it is what we want to compute as the result of the dynamical equation analysis. If we choose at random a different trajectory, with the only condition that the two boundary point remain fixed (a perturbed trajectory ), the chance to obtain exactly the true trajectory will be very small. What characterizes the true trajectory with respect to all possible other perturbed trajectories? Euler contributed to the solution of this problem, but Lagrange developed a complete theory, that was later extended by Hamilton. The true trajectory is the one that minimizes the integral of the vis-viva (i.e., twice the kinetic energy) of the entire motion between Q 1 and Q 2 . This integral is called action and has a constant and well defined value for each perturbed trajectory having constant E ( E depends only on the initial state). Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 4 / 54
The least action principle states that the nature “chooses”, among the infinite number of trajectories starting in q ( t 1 ) and ending in q ( t 2 ), the trajectory that minimizes the definite integral � t 2 C ∗ ( q ( t ) , ˙ S = q ( t )) d t t 1 of a particular state function C ∗ ( q ( t ) , ˙ q ( t )). It is necessary to compute the trajectory in the space Q that minimizes S . The integral between the initial time t 1 and the final time t 2 must obey to the boundary constraints the two time instants. Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 5 / 54
The minimization of a functional is based on a particular mathematical technique, called calculus of variations . A functional is a mapping between a function and a real number; the function shall be considered as a whole, i.e., not a single particular value; in this sense a functional is often the integral of the function. The conditions that guarantee the minimization of S provide a set of differential equations that contain the first and second time derivatives of the q i ( t ); this set completely describes the system dynamics . Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 6 / 54
The differential equations specify the evolution of a physical quantity as the result of infinitesimal increments of time or position; summing up this infinitesimal variations we obtain the physical variables at every instant, knowing only their initial value and possibly some initial derivative: we can say that the motion has a local representation. The action characterizes the motion dynamics requiring only the knowledge of the states at the initial and final times; every intermediate value of the variables can be determined by the minimization of the action, that is a global , rather than a local, measure. The Lagrange approach is based on the definition of two scalar quantities, namely the total kinetic co-energy and the total potential energy associated with the body. The reason for using the term co-energy instead of the term energy , will be clarified later. Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 7 / 54
Lagrangian approach The Lagrange method allows to define a set of Lagrange equations , that have some advantages with respect to the vector equations provided by the Newton-Euler approach. The approach provides n second-order scalar differential equations, directly expressed in the generalized coordinates ˙ q i ( t ) e q i ( t ). If holonomic constraint are present, the constraint force do not appear in the equations. The kinetic co-energies and the potential energies are independent of the reference frame used to represent the body motion. The kinetic co-energies and the potential energies are additive scalars: in a multi-body system the total energies/co-energies are the sum of each energy/co-energy component. Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 8 / 54
Kinetic energy and co-energy for single point-mass The mechanical kinetic energy associated to a point-mass m is defined as the work necessary to increase the linear or angular momentum from 0 to h , i.e., � h C ( h ) = d W 0 where the symbol · indicates the scalar product. The infinitesimal work associated to the mass is given by d W = f · d r where f is the resultant of the applied forces on the mass and d r is the infinitesimal displacement increment and f = d h d t Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 9 / 54
The resulting infinitesimal work is therefore d W = f · d r = d h d t · d r = d h d t · v d t = v · d h and we can write � h C ( h ) = v · d h 0 The kinetic energy is a scalar state function associated to the particle states v and h . Another state function associated to the point-mass, called mechanical kinetic co-energy , is defined as � v C ∗ ( v ) = h · d v 0 As shown in Figure, between the mechanical energy and the co-energy a relation exists C ∗ ( v ) = h · v − C ( h ) Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 10 / 54
This relation is an example of the Legendre transformation Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 11 / 54
In particular, if the mass particle is moving at a velocity significantly smaller that the speed of light c , i.e., it is not a relativistic mass , the relation is h = m v with m constant, and the two “energies” become � h m h · d h = 1 1 2 m h · h = 1 2 m � h � 2 C ( h ) = 0 � v m v · d v = 1 2 m v · v = 1 2 m � v � 2 C ∗ ( v ) = 0 As one can see, in this case the kinetic energy and co-energy are the same since � h � 2 = m 2 � v � 2 This does not happen for relativistic masses where m = m ( v ( t )). Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 12 / 54
In an extended body composed by N masses m i the kinetic co-energy is the sum of the kinetic co-energy of each mass N C ∗ ( v ) = 1 � m i v i · v i 2 i =1 Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 13 / 54
We consider the velocity v i = ˙ r i with respect to R 0 . The velocities in R 0 can be computed from the general relation 0 0 r 1 ( t ) + ˙ 01 ( t ) × ρ 0 ( t ) + ˙ r 0 ( t ) = ω 0 01 ( t ) × ρ 0 ( t ) + R 0 1 ( t ) = ω 0 ˙ 1 ˙ d d 1 ( t ) where the term R 0 b ˙ r i ( t ) is zero, since the point-masses are fixed with respect to the body-frame. Now we consider a purely translatory motion and then a purely rotational motion. Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 14 / 54
Translational motion If the motion is purely translational 0 r 0 i ( t ) = ˙ ˙ d b ( t ) ≡ v 0 ( t ) where v 0 is the total linear velocity with respect to R 0 . All point-masses m i have the same velocity v 0 N C ∗ = 1 m i = 1 2 m tot v 0 · v 0 = 1 2 m tot � v 0 � 2 = 1 � 2 v T 2 v 0 · v 0 0 ( m tot I ) v 0 i =1 where the mass m tot is the total body mass. The kinetic co-energy is equivalent to that of one particle with total mass m tot with the translational velocity v 0 . The total mass m tot can be ideally concentrated in the body center-of-mass C , whose position is r c m i � � r c m tot = → r c = r i m i r i m tot i i Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 15 / 54
Since the velocity is equal for all points of the body, v 0 is also the velocity of the center-of-mass C ; if we use the symbol v 0 c ≡ v 0 for this velocity, we can write C ∗ = 1 2 m tot v 0 c · v 0 c = 1 2 m tot � v 0 c � 2 = 1 2 v T 0 c ( m tot I ) v 0 c that gives the usual rule: “the kinetic energy is half the product of the total mass for the total velocity squared.” Basilio Bona (DAUIN - Politecnico di Torino) Energy October 2013 16 / 54
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